arXiv subjects

Andreas Brandhuber

Publications and source records attributed to Andreas Brandhuber.

At least 19 recordsLinked to original sources

arXiv2026

$\mathcal{N}=4$ single-minus superamplitudes and dual superconformal symmetry

We construct the $\mathcal{N}=4$ supersymmetric completion of the recently proposed single-minus gluon amplitudes in $(2,2)$ signature, which are nonvanishing for all multiplicities on a half-collinear kinematic locus. The superamplitude factorises into a permutation-invariant measure $Δ^{(n-1)}$ with uniform little-group weight that imposes the half-collinearity constraint, a piecewise constant stripped amplitude $\tilde{A}_{1\ldots n}$ that is helicity blind and dual conformal invariant, and (super)momentum conservation delta functions. For $n=3$, our superamplitude reduces to the known $\overline{\rm MHV}$ superamplitude. We prove dual superconformal covariance of the $n$-point superamplitude, and further analyse the $\mathrm{Gr}(k,n)$ Grassmannian integral at $k=1$. Finally, we present the corresponding single-minus superamplitude in $\mathcal{N}=8$ supergravity.

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arXiv2025

The Magnus expansion in relativistic quantum field theory

We investigate the Magnus expansion of the $N$-operator in relativistic quantum field theory, which is related to the $S$-matrix via $S = e^{iN}$. We develop direct methods to compute matrix elements of the $N$-operator, which we refer to as Magnus amplitudes, bypassing scattering amplitudes entirely. At tree level, Magnus amplitudes are expressed in terms of retarded and advanced propagators, with each diagram weighted by factors that we identify as Murua coefficients. At loop level this structure is augmented by the Hadamard cut function, and we establish remarkable relations between loop- and tree-level Magnus amplitudes. Among these, we find that $n$-point one-loop Magnus amplitudes are entirely determined by phase-space integrals of forward limits of $(n{+}2)$-point tree-level amplitudes, and hence related to Murua coefficients, and we generalise this to a class of higher-loop contributions. Furthermore, in the case of heavy particles interacting via massless mediators, we conjecture that Magnus diagrams that contribute to the classical limit are always given by forward limits of trees, and we show this explicitly in a one-loop example. We derive these results studying theories of scalar fields with cubic interactions, but our methods are applicable to general theories as well as to integral functions appearing in gravitational-wave computations. Given that Magnus amplitudes are free of hyper-classical terms, and the known relations between Magnus amplitudes and the radial action, our results lay the groundwork for systematic and efficient calculations of classical observables from quantum field theory.

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arXiv2024

Spinning quadrupoles in effective field theories of gravity

We study the effect of the two independent parity-even cubic interactions $I_1 = {R^{αβ}}_{μν} {R^{μν}}_{ρσ} {R^{ρσ}}_{αβ}$ and $ G_3 = I_1 -2 {R^{μνα}}_β{R^{βγ}}_{νσ} {R^σ}_{μγα}$ on the spectrum of gravitational waves emitted in the quasi-circular inspiral phase of the merger of two spinning objects. Focusing on the aligned spin configuration, we extract the corrections to Newton's potential at linear order in the perturbations, using the four-point amplitude of the massive spinning objects evaluated in the Post-Minkowskian expansion. We then derive the modifications to the quadrupole moments at leading order in the cubic perturbations, using a five-point amplitude with emission of a soft graviton. These modified moments, along with the corresponding potentials, are then employed to calculate the power emitted by gravitational waves during the inspiral phase. Using these results, we determine the changes to the waveforms, up to linear order in spin, in the Stationary Phase Approximation. Finally, we comment on the relation between cubic and tidal perturbations.

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arXiv2024

One-loop Gravitational Bremsstrahlung and Waveforms from a Heavy-Mass Effective Field Theory

Using a heavy-mass effective field theory (HEFT), we study gravitational-wave emission in the scattering of two spinless black holes or neutron stars of arbitrary masses at next-to-leading order in the Post-Minkowskian expansion. We compute the contributions to the one-loop scattering amplitude with four scalars and one graviton which are relevant to the calculation of the waveforms, also presenting expressions of classical tree-level amplitudes with four scalars and up to two radiated gravitons. The latter are obtained using a novel on-shell recursion relation for classical amplitudes with four scalars and an arbitrary number of gravitons. Our one-loop five-point amplitude is expressed in terms of a single family of master integrals with the principal value prescription for linearised massive propagators, which we evaluate using differential equations. In our HEFT approach all hyper-classical iterations and quantum corrections to the amplitude are dropped at the diagrammatic level, thereby computing directly contributions to classical physics. Our result exhibits the expected factorisation of infrared divergences, the correct soft limits, and highly nontrivial cancellations of spurious poles. Finally, using our amplitude result we compute numerically the corresponding next-to-leading corrections to the spectral waveforms and the far-field time-domain waveforms using the Newman-Penrose scalar $Ψ_4$.

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arXiv2024

Spinning binary dynamics in cubic effective field theories of gravity

We study the binary dynamics of two Kerr black holes with arbitrary spin vectors in the presence of parity-even and parity-odd cubic deformations of gravity. We first derive the tree-level Compton amplitudes for a Kerr black hole in cubic gravity, which we then use to compute the two-to-two amplitudes of the massive bodies to leading order in the deformation and the post-Minkowskian expansion. The required one-loop computations are performed using the leading singularity approach as well as the heavy-mass effective field theory (HEFT) approach. These amplitudes are then used to compute the leading-order momentum and spin kick in cubic gravity in the KMOC formalism. Our results are valid for generic masses and spin vectors, and include all the independent parity-even and parity-odd cubic deformations of Einstein-Hilbert gravity. We also present spin-expanded expressions for the momentum and spin kicks, and the all-order in spin deflection angle in the case of aligned spins.

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arXiv2024

Spinning waveforms in cubic effective field theories of gravity

We derive analytic all-order-in-spin expressions for the leading-order time-domain waveforms generated in the scattering of two Kerr black holes with arbitrary masses and spin vectors in the presence of all independent cubic deformations of Einstein-Hilbert gravity. These are the two parity-even interactions $I_1$ and $G_3$, and the parity-odd ones $\tilde{I}_1$ and $\tilde{G}_3$. Our results are obtained using three independent methods: a particularly efficient direct integration and tensor reduction approach; integration by parts combined with the method of differential equations; and finally a residue computation. For the case of the $G_3$ and $\tilde{G}_3$ deformations we can express the spinning waveform in terms of the scalar waveform with appropriately shifted impact parameters, which are reminiscent of Newman-Janis shifts. For $I_1$ and $\tilde{I}_1$ similar shifts occur, but are accompanied by additional contributions that cannot be captured by simply shifting the scalar $I_1$ and $\tilde{I}_1$ waveforms. We also show the absence of leading-order corrections to gravitational memory. Our analytic results are notably compact, and we compare the effectiveness of the three methods used to obtain them. We also briefly comment on the magnitude of the corrections to observables due to cubic deformations.

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arXiv2024

Resummed spinning waveforms from five-point amplitudes

We compute the classical tree-level five-point amplitude for the two-to-two scattering of spinning celestial objects with the emission of a graviton. Using this five-point amplitude, we then turn to the computation of the leading-order time-domain gravitational waveform. The method we describe is suitable for arbitrary values of classical spin of Kerr black holes and does not require any expansion in powers of the spin. In this paper we illustrate it in the simpler case of the scattering of one Kerr and one Schwarzschild black hole. An important ingredient of our calculation is a novel form of the Compton amplitude with spinning particles including contact terms derived from matching to black-hole perturbation theory calculations. This ensures that our waveform is valid up to at least fourth order in the spin. Our method can be applied immediately to generate improved waveforms once higher-order contact terms in the Compton amplitude become available. Finally, we show the formula for the gravitational memory to all orders in the spin, which is in agreement with our results.

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arXiv2023

Amplitudes, Hopf algebras and the colour-kinematics duality

It was recently proposed that the kinematic algebra featuring in the colour-kinematics duality for scattering amplitudes in heavy-mass effective field theory (HEFT) and Yang-Mills theory is a quasi-shuffle Hopf algebra. The associated fusion product determines the structure of the Bern-Carrasco-Johansson (BCJ) numerators, which are manifestly gauge invariant and with poles corresponding to heavy-particle exchange. In this work we explore the deep connections between the quasi-shuffle algebra and general physical properties of the scattering amplitudes. First, after proving the double-copy form for gravitational HEFT amplitudes, we show that the coproducts of the kinematic algebra are in correspondence with factorisations of BCJ numerators on massive poles. We then study an extension of the standard quasi-shuffle Hopf algebra to a non-abelian version describing BCJ numerators with all possible gluon orderings. This is achieved by tensoring the original algebra with a particular Hopf algebra of orderings. In this extended version, a specific choice of the coproduct in the algebra of orderings leads to an antipode in the resulting Hopf algebra that has the interpretation of reversing the gluons' order within each BCJ numerator.

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arXiv2023

The SAGEX Review on Scattering Amplitudes, Chapter 1: Modern Fundamentals of Amplitudes

This chapter introduces the foundational elements of scattering amplitudes. It is meant to be accessible to readers with only a basic understanding of quantum field theory. Topics covered include: the four-dimensional spinor-helicity formalism and the colour decomposition of Yang-Mills scattering amplitudes; the study of soft and collinear limits of Yang-Mills and gravity amplitudes; the BCFW recursion relation and generalised unitarity, also in the superamplitudes formalism of $\mathcal{N}{=}4$ supersymmetric Yang-Mills; an overview of standard and hidden symmetries of the $S$-matrix of $\mathcal{N}{=}4$ supersymmetric Yang-Mills, such as the conformal, dual conformal and Yangian symmetries; and a brief excursus on form factors of protected and non-protected operators in Yang-Mills theory. Several examples and explicit calculations are also provided.

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arXiv2023

The SAGEX Review on Scattering Amplitudes

This is an introduction to, and invitation to read, a series of review articles on scattering amplitudes in gauge theory, gravity, and superstring theory. Our aim is to provide an overview of the field, from basic aspects to a selection of current (2022) research and developments.

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arXiv2022

Kinematic Hopf Algebra for BCJ Numerators in Heavy-Mass Effective Field Theory and Yang-Mills Theory

We present a closed formula for all Bern-Carrasco-Johansson (BCJ) numerators describing $D$-dimensional tree-level scattering amplitudes in a heavy-mass effective field theory with two massive particles and an arbitrary number of gluons. The corresponding gravitational amplitudes obtained via the double copy directly enter the computation of black-hole scattering and gravitational-wave emission. Our construction is based on finding a kinematic algebra for the numerators, which we relate to a quasi-shuffle Hopf algebra. The BCJ numerators thus obtained have a compact form and intriguing features: gauge invariance is manifest, locality is respected for massless exchange, and they contain poles corresponding to massive exchange. Counting the number of terms in a BCJ numerator for $n{-}2$ gluons gives the Fubini numbers $\mathsf{F}_{n-3}$, reflecting the underlying quasi-shuffle Hopf algebra structure. Finally, by considering an appropriate factorisation limit, the massive particles decouple, and we thus obtain a kinematic algebra and all tree-level BCJ numerators for $D$-dimensional pure Yang-Mills theory.

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arXiv2021

Celestial Superamplitudes

We study celestial amplitudes in (super) Yang-Mills theory using a parameterisation of the spinor helicity variables where their overall phase is not fixed by the little group action. In this approach the spin constraint $h-\bar{h}=J$ for celestial conformal primaries emerges naturally from a new Mellin transform, and the action of conformal transformations on celestial amplitudes is derived. Applying this approach to $\mathcal{N}\!=\!4$ super Yang-Mills, we show how the appropriate definition of on-shell superspace coordinates leads naturally to a formulation of chiral celestial superamplitudes and a representation of the generators of the four-dimensional superconformal algebra on the celestial sphere, which by construction annihilate all tree-level celestial superamplitudes.

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arXiv2021

Classical gravitational scattering from a gauge-invariant double copy

We propose a method to compute the scattering angle for classical black hole scattering directly from two massive particle irreducible diagrams in a heavy-mass effective field theory approach to general relativity, without the need of subtracting iteration terms. The amplitudes in this effective theory are constructed using a recently proposed novel colour-kinematic/double copy for tree-level two-scalar, multi-graviton amplitudes, where the BCJ numerators are gauge invariant and local with respect to the massless gravitons. These tree amplitudes, together with graviton tree amplitudes, enter the construction of the required $D$-dimensional loop integrands and allow for a direct extraction of contributions relevant for classical physics. In particular the soft/heavy-mass expansions of full integrands is circumvented, and all iterating contributions can be dropped from the get go. We use this method to compute the scattering angle up to third post-Minkowskian order in four dimensions, including radiation reaction contributions, also providing the expression of the corresponding integrand in $D$ dimensions.

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arXiv2021

A new gauge-invariant double copy for heavy-mass effective theory

We propose a new form of the colour-kinematics/double-copy duality for heavy-mass effective field theories, which we apply to construct compact expressions for tree amplitudes with heavy matter particles in Yang-Mills and in gravity to leading order in the mass. In this set-up, the new BCJ numerators are fixed uniquely and directly written in terms of field strengths, making gauge invariance manifest. Furthermore, they are local and automatically satisfy the Jacobi relations and crossing symmetry. We construct these BCJ numerators explicitly up to six particles. We also discuss relations of the BCJ numerators in the heavy-mass effective theory with those in pure Yang-Mills amplitudes.

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arXiv2020

From amplitudes to gravitational radiation with cubic interactions and tidal effects

We study the effect of cubic and tidal interactions on the spectrum of gravitational waves emitted in the inspiral phase of the merger of two non-spinning objects. There are two independent parity-even cubic interaction terms, which we take to be $I_1 = {R^{αβ}}_{μν} {R^{μν}}_{ρσ} {R^{ρσ}}_{αβ}$ and $G_3 = I_1-2 R^α\,_μ\,^β\,_ν R^μ\,_ρ\,^ν\,_σ R^ρ\,_α\,^σ\,_β$. The latter has vanishing pure graviton amplitudes but modifies mixed scalar/graviton amplitudes which are crucial for our study. Working in an effective field theory set-up, we compute the modifications to the quadrupole moment due to $I_1$, $G_3$ and tidal interactions, from which we obtain the power of gravitational waves radiated in the process to first order in the perturbations and leading order in the post-Minkowskian expansion. The $I_1$ predictions are novel, and we find that our results for $G_3$ are related to the known quadrupole corrections arising from tidal perturbations, although the physical origin of the $G_3$ coupling is unrelated to the finite-size effects underlying tidal interactions. We show this by recomputing such tidal corrections and by presenting an explicit field redefinition. In the post-Newtonian expansion our results are complete at leading order, which for the gravitational-wave flux is 5PN for $G_3$ and tidal interactions, and 6PN for $I_1$. Finally, we compute the corresponding modifications to the waveforms.

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arXiv2020

On higher-derivative effects on the gravitational potential and particle bending

Using modern amplitude techniques we compute the leading classical and quantum corrections to the classical gravitational potential between two massive scalars induced by adding an $R^3$ term to Einstein gravity. We then study the scattering of massless scalars, photons and gravitons off a heavy scalar in the presence of the same $R^3$ deformation, and determine the bending angle in the three cases from the non-analytic component of the scattering amplitude. Similarly to the Einstein-Hilbert case, we find that the classical contribution to the bending angle is universal, but unlike that case, universality is preserved also by the first quantum correction. Finally we extend our analysis to include a deformation of the form $ΦR^2$, where $Φ$ is the dilaton, which arises in the low-energy effective action of the bosonic string in addition to the $R^3$ term, and compute its effect on the graviton bending.

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arXiv2020

Eikonal phase matrix, deflection angle and time delay in effective field theories of gravity

The eikonal approximation is an ideal tool to extract classical observables in gauge theory and gravity directly from scattering amplitudes. Here we consider effective theories of gravity where in addition to the Einstein-Hilbert term we include non-minimal couplings of the type $R^3$, $R^4$ and $FFR$. In particular, we study the scattering of gravitons and photons of frequency $ω$ off heavy scalars of mass $m$ in the limit $m\gg ω\gg |\vec{q}\,|$, where $\vec{q}$ is the momentum transfer. The presence of non-minimal couplings induces helicity-flip processes which survive the eikonal limit, thereby promoting the eikonal phase to an eikonal phase matrix. We obtain the latter from the relevant two-to-two helicity amplitudes that we compute up to one-loop order, and confirm that the leading-order terms in $ω$ exponentiate à la Amati, Ciafaloni and Veneziano. From the eigenvalues of the eikonal phase matrix we then extract two physical observables, to 2PM order: the classical deflection angle and Shapiro time delay/advance. Whenever the classical expectation of helicity conservation of the massless scattered particle is violated, i.e. the eigenvalues of the eikonal matrix are non-degenerate, causality violation due to time advance is a generic possibility for small impact parameter. We show that for graviton scattering in the $R^4$ and $FFR$ theories, time advance is circumvented if the couplings of these interactions satisfy certain positivity conditions, while it is unavoidable for graviton scattering in the $R^3$ theory and photon scattering in the $FFR$ theory. The scattering processes we consider mimic the deflection of photons and gravitons off spinless heavy objects such as black~holes.

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arXiv2019

A note on the absence of $R^2$ corrections to Newton's potential

We consider Einstein gravity with the addition of $R^2$ and $R^{μν} R_{μν}$ interactions in the context of effective field theory, and the corresponding scattering amplitudes of gravitons and minimally-coupled heavy scalars. First, we recover the known fact that graviton amplitudes are the same as in Einstein gravity. Then we show that all amplitudes with two heavy scalars and an arbitrary number of gravitons are also not affected by these interactions. We prove this by direct computations, using field redefinitions known from earlier applications in string theory, and with a combination of factorisation and power-counting arguments. Combined with unitarity, these results imply that, in an effective field theory approach, the Newtonian potential receives neither classical nor quantum corrections from terms quadratic in the curvature.

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